Homework 9

NOTE: this assignment will be turned in on APRIL 10 (before the WPR), and count as part of the WPR grade.

In this assignment, you'll be investigating the definition of distance between compact sets. There are a few equivalent ways to define this distance (see Problem 14 in Chapter 2) for another example. This homework assignment requires a review of the definition of continuity, and provides an example of using the Extreme Value Theorem.

Distance Between a Point and a Set

1. Let x be a point in a metric space M, and let $S\subset M$. Prove that the function $f:S\to\mathbb{R}$ defined by $f(y)=d(x,y)$ is continuous. You should use the epsilon-delta definition of continuity. (Hint: You'll need the triangle inequality.)

2. Now suppose that $S\subset M$ is compact and x is still a point in the metric space. Prove that some point $s\in S$ is "closest" to x. In other words, show that there exists some $s\in S$ such that $d(s,x)\leq d(y,x)$ for all $y\in S$. (Hint: Use one of the "big theorems" that we've been discussing in class.)

3. By the previous problem, we can define the distance $d(x,S)$ between a point and a set to be the distance between x and the point in S which is closest to x. When is $d(x,S)=0$?

4. Suppose the metric space is chosen to be $\mathbb{R}$. show that if S is not closed, there may be no minimum distance between a point x and S. What if S is closed but not bounded?

Distance Between Sets

5. Given that the function $f(y)=d(x,y)$ is continuous, as proven in Problem 1, show that the function $g(x)=d(x,S)$ is continuous.

6. Suppose that S and T are compact subsets of M. One can define $g:T\to\mathbb{R}$ by $g(x)=d(x,S)$. Prove that g obtains a minimum value. What are the implications for the sets S and T?

7. Hence, we can define $d(S,T)$. Show that $diam(S\cup T)\leq diam(S)+diam(T)+d(S,T)$. Provide an example (easiest in $\mathbb{R}$) showing that equality is possible.

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